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Harder to C++: Monads for Mortals [7], Monad Composition

Early authors on monads, e.g. [Moggi], [Wadler], describe several types of monads, notably for side-effects, state, exceptions and IO. Of course we want to write programs that have all these features. There are, in principle, two possible approaches. First: write a separate monad each time we think up a new piece of non-pure-functional … (shall we dare call it functionality?). Second: compose different standard monads into larger wholes? In [part 2] of his excellent tutorial on monads Mike Vanier writes:

‘ Functional programmers talk about “composability” all the time, with the implication that if some aspect of a programming language isn’t composable, it’s probably not worth much.’

So, clearly composition is key and we will see some monads composed into larger wholes here. To be clear: I don’t mean function composition, I mean monad composition. In C++. To be precise, I will define:

A monad that executes a function which maintains a state (side-effect).

A monad that catches and handles exceptions.

A monad that writes the result of function composition to console or an error message in case an exception occurred.

Then I will compose those monads into a larger whole, with preservation of imperative ‘functionality’. We will see that it actually works. Then I will show a very general imperative C++ function, that has the same functionality, and we will compare size and performance of both approaches.

Monad Composition

We will use a single monad template and instantiate it for a State type, an Exception type, and a std::cout wrapper. The monad template looks like this:

The template consists of a wrapped value and a field to hold state of some sort, e.g. actual state, an exception, or an io stream. The rest of the code is just what is generally referred to as ‘copy control’.

The ‘S’ parameter will be instantiated using three classes: State, Exception, and Cout

Cout is a std::cout wrapper. We need it because io streams cannot be copied or assigned, but by the design of our monad template (S is not a reference or pointer) we need something that actually can be copied or assigned.

For the monad template we define three overloads of the bind function, one for each instantiation of the template:

In the last bind function we see the familiar output operator. In order to make it work, we need overloads of this operator for the monad template, and the three auxiliary classes (or ‘structs’ if you like):

The last thing we need is some function that can be applied by the bind functions. These functions cannot be type agnostic: the return type differs from the parameter type. So we have three overloads of them, and they do exactly the same. Let’s first define some handy type aliases:

After a code exposé of 34 lines (it pays to remember the approximate factor of 6 for later).

Let’s compare the sizes of both approaches. Well, the monad composition is ridiculously much larger than the ordinary solution.

Performance

Performance measurements were set up in a way comparable to measurements in part 6: the code above is run a large number of iterations, and time is measured. This is done for 10 runs, and the average over the times is calculated. Before starting measurements, the monad composition is allowed a warming up run, to prevent an outlier. The default release build of the resulting program was run from a command console.

Now, time performance results are dominated almost completely by the use of std::cout and throwing exceptions. If both are used, or occur, times are comparable. If however, we do not log to the console, and have no exceptions thrown (we set value to 32), that is, we measure the performance at the naked difference of the constructed code, the regular code takes only 1.6% of the time the monad solution takes, i.e. the regular solution is about 60 times as fast, see the image below for a typical performance run.

in this particular case the ordinary C++ function is 38.8 / 0.64 = 60.6 times as fast. The time it takes the regular function to execute is 100% x 0.64 / 38.8 = 1.6% of the time it takes the monad composition to execute. Please note that the variation in measured times is limited.

Conclusions

Let’s be brief. The code size and time performance of the monad composition are not in sensible proportion to size and performance of the regular function: 6 : 1 and 60 : 1 respectively. The code size of monad composition is ridiculous, monad composition kills C++ performance.

Wrap Up of Sequential Monads

We are a now at the end of 7 installments experimenting with monads, so let’s wrap up what we have so far. The question is: “Can monads be useful in day-to-day C++ programming?”. And the answer is “No”.

We have looked briefly at a C++ template meta programming (which constitutes a simple functional language) approach. The disadvantages are:

Complex syntax, hence diminished simplicity and clarity (one day your code will be maintained by another developer!)

Although it is possible to use this approach, the language does not support it. So it is unreasonably hard. (free after Stroustrup’s argument that OO development is possible in C, or assembly, but the language… )

There are no debugging facilities, which makes it very hard to develop substantial amounts of code as meta programs.

We have seen a simple and short implementation of the monad in mainstream C++. However, this implementations has the drawback that its performance dies if it is applied to large data structures.

The drawback could be remedied by replacing the unit function by a complete copy control set for the monadic template.

Lazy monads can be elegantly implemented as well, but they have no performance whatsoever, due to the fact that code is evaluated twice, once to create the expression to evaluate, and another time to indeed evaluate the expression.

Furthermore, we have seen, in this episode that if we want to write somewhat larger programs using monad composition, we get very large programs without performance.

I think I’m done now with at least the sequential monad. I have no further interest. It is tedious, needlessly complicated, and leading to inefficient results when trying to construct meaningful programs from different monads in C++.

I wrote ‘sequential monad’ above, because there is still the continuation monad to investigate; the suggested solution to Callback Hell in concurrency contexts. So, for starters, what exactly is Callback Hell? We will see in part 8.

In parts 2 – 5 we have seen four different implementations of the monad. Now, it is time for a shootout. I want to see which version has best time performance and compare it with a regular task to evaluate the cost of using a monad.

The monad implementations are:

The first implementation that copies all its data.

The lazy implementation that evaluates at explicit call.

The references and move semantics implementation.

The pointer based implementation.

The Task at Hand

The challenge is to create a task that is hard to optimize by cleverly removing parts of the functionality. Imagine we have an algorithm that fills a vector with arbitrary data, which is then thrown away unused because the vector goes out of scope. We wouldn’t want the compiler to decide that it is useless to create the vector altogether and skip the code.

Further, I will make the task less predictable by extensive use of the rand() function. I know, rand() is not a great pseudo random engine. However, rand() is not used here to simulate randomness, but to get values that are available only at runtime, not at compile time – when the code optimizer runs. This way, the code cannot have optimizations based on compile time knowledge of constant values.

The algorithm that will be implemented by the functions that are ‘applied’ by the monad, and that will also be implemented by two regular functions is as follows (we will be working with the ValueObject class used in earlier episodes, but with a method added that retrieves a pointer to the payload):

Receive a (pointer / reference to a) ValueObject object.

Retrieve a char from the ValueObject object’s payload, with a random index, stored it in a global data structure.

Create a size for a new ValueObject object, that depends on the size of the received object, and the pseudo random generator.

Main Routine

In the main routine, we have a composition of 25 applications of the onemore function by the bind function. We execute this composition iterations times, which we call a run. We measure the time of each run. We average over the runs, and print the result tot screen.

The number of iterations and runs are given by rand(), which is seeded with a number obtained from the user.

Scoping is such that all objects are created and destroyed in measured time.

This is the code for the reference and move semantics based implementation:

As earlier, the timer code is from Simon Wybranski. The outer braces are just to assure different implementations do not interfere.

The iterations for the lazy monad differ from the above articulation in that it also makes a call: m2() to evaluate the constructed expression.

Regular Algorithms

The monad implementations are compared to regular algorithms that are functionally equivalent, but do not use monads. We will use one variant with references, and one variant with pointers and heap allocation.

The algorithm used is the same as above. The call in the main loop is terrible:

Procedure

Time performance was measured using a release build, with default compiler switches for release builds, and the runs were executed outside of Visual Studio. That is, the program was started from a console window, not from Visual Studio.

Results

And now the results:

So, what do we see?

The monad implementation based on references and move semantics (yellow) is the fastest. But the pointer based implementation has about the same performance.

The regular implementations are only slightly faster (red, orange).

The lazy monad (blue) is very slow, as expected.

Conclusions

The choice is clear. In following episodes, we will work with the monad built on references and move semantics. If required, we might use the pointer based variant as well. I will not use the lazy monad, unless it is clear that very, very special circumstances make it a favorable choice.

I was surprised by the small difference (if at all) between the fast monad implementation and the regular algorithms; the monad covers about twice the source code as the regular function – it also includes the bind function. So (hypothesis), the cost of using the references and move semantics monad is negligible.

Next

Next time we will start looking at various types of monads and how we could build compositions of them.

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Harder to C++: Monads for Mortals [5], Pointers

In part 2 we have seen a small and elegant implementation of the monad. In part 4 we have seen how references and move semantics can be used to make the monad’s performance independent of the size of the wrapped value. In this part we will see another implementation the monad based on a pointer.

Implementation of a Monad with a Smart Pointer

We have seen in part 2 that the type constructor is represented by a template in C++. In part 2 this is a simple struct, in part 3 it is a std::function holding a lambda expression – so as to implement delayed evaluation, and in this part, it is a smart pointer, the std::unique_ptr to be precise.

I think the most important result here is what we don’t see: calls to the move constructor and the accompanying call to the destructor of the ValueObject.

Pro’s, Con’s, and Questions

The above image of the output is nicely constrained. Functions return the unique_ptr by value, i.e. it gets copied which is ok, and the bind function takes a reference to a unique_ptr to elicit the raw pointer from. We see the calls to ValueObject destructor occur after assignment to m2, and when leaving the anonymous scope.

So, pro’s for this implementation are its simplicity, clarity, and efficiency: the overhead of a unique_ptr is comparable to the overhead of a raw pointer. Values are not copied, so performance is size independent.

You may wonder, though, whether it is a good idea to create all the monad’s values on the heap, which is not so efficient. In part 6 we take a look at what we have so far and see what works best.

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Harder to C++: Monads for Mortals [4], References and Move Semantics

In part 3 we have seen a simple monad with delayed evaluation. Until we find a use for delayed evaluation that adds value to mainstream imperative practices, we will focus on the eager or strict evaluating monad. Until the shootouts, for sure.

In part 2 we have seen a simple implementation of the eager monad. A drawback of that specific implementation is that it copies its value around, which will lead to poor performance in case of large data structures.

In this part we will examine an improved implementation that has the same time performance for any data size (that fits in RAM memory). We will improve the design of part 1 by using references and move semantics. To me, a good read on move semantics seems this article on StackOverflow.

Of course, we want to see the move semantics in action, so we can be sure the presented code indeed uses move semantics and references, hence is a performant solution. To be able to trace how an object, representing the monad value is moved around, we construct a class that will serve as the value, and which contains all necessary constructors, destructors and assignment operators, each fitted with an output statement reporting its execution.

We will add the same trace instruments to the monad template,. This requires that the unit function is integrated into the monad template in the form of various constructors (hence destructors) and assignment operators. To integrate the unit function into the monad template this way is not different from the practice Eric Lippert exercises in his tutorial on monads. There will be no use of a so called swap function in applicable constructors and assignment operators; that would again merge what we are trying to separate here.

Implementation of a Monad with References and Move Semantics

The Value Class

We will use the following class as the value in the monad to evaluate the use of refs and move semantics.

The constructors, destructor and assignment operators are all textbook examples. The class has a payload which we will use in performance measurements. It is a char array of size sz. The ValueObject class has a method (at the bottom) that returns size sz.

Most implementations are standard. Note that there is no parameterless default constructor. The signature of the first constructor is special in that one would expect a const V& parameter, like so

monad(const V& v) : value(std::move(v))

But doing that would coerce the use of the ValueObject’s copy constructor instead of the desired move constructor; the copy constructor has a const parameter, whereas the move constructor has a non-const parameter.

As with the ValueObject class, each method writes a line to the console to mark that it has been executed.

We test the monad template with code that exercises all methods above.

For use with the ValueObject, we will add two example functions. In analogy to previous installments, the first function divides the size of the payload by 2, the second function tells us if the size is even.

The good news is: the code indeed only uses references and move semantics.

But OMG, What A Shocking Lot Of Function Calls! What a Hydrocephalic Overhead!!!

Look what happens between entering the function divideby2 and the subsequent call to the bind function: No less than 8 calls to constructors and destructors. What a shocking waste of time and effort. To create 2 objects, apply 2 functions, and then clean up the mess, 24 function calls are required. What circumstances could ever justify such an overhead?

But of course, we cannot blame just the monad for this, it is also the move semantics. And that is what really shocks me. How could they do that? Now I understand why the committee is ready to embrace the monad. Who cares about the overhead of the monad if you already have the overhead of move semantics?

Minimal Verification that Time Performance is Independent of Size

Ok, let’s put the drama behind us and see if this solution is indeed size independent. We will compare a payload of size 1 to a payload of 10,000,000,000.

To make the test size independent we will use an identity function. This function does not modify the data, it just moves the data on – Yes it is still immutable data. The identity function will not muddle time measurements with memory (de-)allocations for the actual data, so we will only measure the overhead for the monad and the move semantics.

Comparable values (about 2 ms) are indeed obtained for both the large and the small payload.

Pros, Cons, and Questions

So, a strict monad that is based on references and move semantics takes 2 ms to do nothing 30,000 times, independently of size of the data structure :-) .

The amount of time per function application is small, so we might conclude that references and move semantics indeed provide an advantage over code that always copies its data structures, as we had in parts 1 and 2.

On the other hand, I find the amount of constructor and destructor calls distressing: returning a monad from a function takes two constructor calls and two destructor calls. Can we do better than that? Is it worth the trouble to try and do better than that?

I think it surely is worth the trouble. There is always the potential comparison between 2 programs that are functionally equivalent, but one program needs only half the instructions the other program needs. The advantages of the smaller program are of great importance. It can run equally fast on much less expensive hardware, it can save 50% battery power compared to the other program. On equal hardware it leaves room for the execution of other instructions that can contribute to a significantly better UX. So, finding the best performance brings you a competitive edge that can make a world of differences in the long run, or at large scale.

Next episode we will investigate if a pointer based monad can do any better than the constructs presented here.

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Harder to C++: Monads for Mortals [3], Lazy implementation

This blog post is part 3 in a series. This part presents a simple implementation of a lazy evaluating monad. It is a minimal variation of the implementation in part 2.

Below we will see how the implementation is constructed, then an example is given that provides initial verification that the construction is indeed a monad.

Implementation of a Lazy Monad

Implementation starts again with some includes and a using statement. The first important construct is the monad type constructor, a C++ template which in this particular case takes the form of a std::function template:

Of course, a suitably formed pointer to function object would suffice as well. The std::function template has the advantage that it looks neat, and we can instantiate it in an elegant way using a lambda expression. Thus upholding some of the functional character of the monad’s roots.

The unit function, which creates instances of the ‘monadic’ function object is as follows:

The returned function is not evaluated before function call is applied to it. Hence using this bind operator we can construct long and complex function compositions, which get evaluated only after we explicitly do so.

The above triple is the complete implementation of the monad. Yes I know, there should be two additional operations defined as well: the sequencing operator and the fail operator (see e.g. Mike Vanier, part 3. However, many texts discussing monads indicate that these operations are uninteresting. So, we will skip them in this series.

Let’s add the same functions as in part 2, and run a simple composition:

Note that the functions ‘divideby2′ and ‘even’ have not changed. The only change in the _tmain function is that in the output statement m2.value has been replaced by m2(). Note the parentheses, which denote lazy evaluation of the constructed composition.

Minimal Verification that Monadic Laws Hold

The minimal verification is the lazy analog to verification in the eager case. It looks like this:

Pros, Cons

‘Lazy’ allows he evaluation of lists that are unbounded (endless, infinite)

Delayed evaluation of function compositions allows for an optimization before evaluation of the composition takes place.

What to think of this?

The first advantage strikes me as odd; I usually do not write programs that make calculations which’ results are not required.

The second advantage is odd as well. My computer is as powerful as a Turing machine, hence it can process input lists of unboundedly length. Without lazy evaluation.

The third advantage could be a real advantage, but I wouldn’t know how to do this in C++; the advantage presupposes that it is possible to create code at runtime and then compile and execute it, which isn’t true for C++.

I do see a disadvantage, though, and we will get to it in detail when doing some shootouts between the various alternative monad implementations. The disadvantage is that the software does extra work. It first creates an expression, then evaluates it. So in C++, for each lazy evaluating program, there is an equivalent eager evaluating program that is faster and smaller, hence more competitive.

In the next part of this series we will work on an (eager) implementation that has the same time performance characteristics for any size value. There will be many references and move semantics. Hmm … mjummy ;-)

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Harder to C++: Monads for Mortals [2], First implementation

This blog post is part 2 in a series. Part 1 is a general introduction. This part discusses a first implementation of the monad in C++.

There are two fundamentally different approaches to implementing monads in C++. Let’s first discuss the approach we will not pursue.

A Way We Will Not to Implement a Monad in C++

The C++ template mechanism is a full functional language (this discovery, by Erwin Unruh, was a bit of a surprise). As a consequence C++ templates are sometimes used for template meta programming: programs can be written using (only) C++ templates and are as such evaluated / executed, at compile time. There are not many practical uses for template meta programming. One practical use was to unroll (small) loops in order to gain performance. This gain, however, can nowadays also be obtained by modern, optimizing C++ compilers.

The discovery that the C++ template mechanism constitutes a functional language, also has lead some people to implement monads in a template meta programming based functional language. Examples are the Cee Plus Plus Monad Example, and Implementing Monads for C++ Template Metaprograms. However, these implementations live in a part of C++ that is little used. Moreover, the required syntax is complex, which makes it unattractive for general use.

The above approach and the goal pursued here are, I think, trying to answer different questions. The above examples seem to try to mimic functional programming with C++, and implement monads in that context. Contrary to bringing C++ to functional programming, I want to know if bringing the functional programming construct ‘monad’ to C++ will provide me with an advantage, and to what cost.

Below we will start with a simple implementation of a monad in mainstream C++ and show that it is indeed a monad. Then we will see what we like or dislike. Note that we will not be doing any functional programming, which, in fact, is quite in the spirit of Eugenio Moggi’s intentions.

A Simple, Mainstream C++ Implementation of the Monad

Implementation starts with some includes and a using statement. The first important construct is the monad type constructor, or rather: the C++ template.

We saw in part 1 that the result of the unit function has the semantics of a computation. As you can see, this C++ class / struct template does not have any computations, it is a trivial computation. For one thing, we could add a function application operator to it ( operator() ), that would perform some non trivial computation, and we might indeed at some point do so.

It is simplicity itself. An alternative could have been to implement the unit function as the monad struct constructor. That would have been more efficient, so we will do that in a number of following implementations.

I have given the bind function the form of an overload of the bitwise OR operator ” | “.

I got the idea to use an operator instead of a function called ‘bind’ from the CeePlusPlus example referred to above. However, that code overloads the ” >>= ” operator, in order to make the code resemble Haskell somewhat. I prefer the ” | ” because it is the same symbol as the Unix / Linux shell pipe operator. And indeed, we will see shortly that in function composition, bind can well be considered to pipe results from function to function.

The above bind function first ‘unwraps the value’, by referencing the value of the monad object, it then applies the function ‘func’ to it, and returns the result wrapped in a new monad template object.

At this point we have a first implementation of a monad! Note how very small and simple it is. Let’s add some functions and give it a spin.

Why is the bind operator not a member of the monadic template like with Lippert? Note that doing so would turn the monadic template into the complete monad. Well, this is C++, and I like to follow C++ best practices as accumulated by Sutter and Alexandrescu here, see item 44 “Prefer writing nonmember nonfriend functions”. There are more reasons, having to do with composition of different types of monads. But that’s for later…

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Harder to C++: Monads for Mortals [1]

The monad is a programming construct, a coding idiom, or perhaps even a design pattern. Its roots lie in Category Theory, coming to programming via programming language semantics and functional languages. I have been reading up on monads (there are multiple, closely related variants), specifically monads for C++. It turns out that the concept of a monad is quit slippery; one minute you think you have a nice understanding, the next minute you are back at zero.

It seems to me that for a thorough understanding of what monads are, you need some knowledge of Category Theory and neighboring fields in logics and algebra, and sound knowledge of the functional programming language Haskell. But alas, I’m just a simple mortal programmer. Moreover:

I don’t want to know about Category Theory!

I don’t care about have time for Haskell !

I just want to write programs in C++!

An alternative route to understanding monads that suits me (simple mortal soul) better, then, is to experiment with various implementations of monads in C++, and evaluate the results, experiment with an improved implementation. And so on.

I have been busy running this experimental cycle, and will describe the condensed experiences in a number of blog posts.

Many interesting things happened, but I can’t say I found a satisfying implementation. Indeed, I have come to doubt the usefulness of monads for application in C++, despite its conceptual and syntactical beauty. Even worse, I have come to doubt the usefulness of some other additions to the C++ language, new since C++11.

But monads are hot!, They must be useful in C++! Well, let us be clear about our motive to write programs in C++. The unique selling points of C++ are (i) the fastest code, against (ii) the smallest footprint. I think people buy in on C++ for exactly these selling points.

Let’s compare with driving a car (what a boyish thing to do). Programming in C++ is then, of course, driving a formula 1 race car; fast, small, unsafe, and uncomfortable. You want comfort, drive a large American saloon, i.e. code in a language designed for developer productivity. You want to be safe, correct? Drive a Toyota Prius, i.e. program a functional language.

Of course, Formula 1 drivers want to be as safe and comfortable as possible. But not at the expense of performance!

So, the monad can be a contribution to C++, if(f) it has no time, space performance cost.

Some Background on Monads

Let’s see what monads are, where they come from, and how they got into programming.

For a C++ developer, a monad is a triple <M, u, b> where:

M is a template (yes, m for monad :-) ).

u is a factory function, in general called unitof signature a -> M<a>, that given an object or value of some type a creates a template object of type M<a>.

Applies a function f: a -> M<b> (from type a to a template of type M<b>)

Returns a template of type M<b>

This triple has to satisfy three laws in order to be a monad:

Left identity: bind(M<a>, u) = M<a> (so left means left argument for bind)

Right identity: for an f: a->M<b>, bind(M<a>, f) = f(a)

Associativity: bind(x, f); bind(y, g) = bind(x, f; bind(y, g))

In the description of the associative law:

x is a variable of type M<a>

y is the result of bind(x, f)

the ‘;’ operator denotes left to right sequencing of operations

f is a function of type a -> M<b> and g is a function of type b -> M<c>

Now. let’s add some intuition.

According to Eugenio Moggi, the semantics of M is that of a computation. He means to include a general class of computations, e.g. partial computations, nondeterministic computations, side-effects, exceptions, continuations, and interactive input and output, and possibly more.

Further, u is the inclusion of value a in a computation, as a parameter. So, we may think of M as a computation.

The application function b (applied to function f) is the extension of a function f from values to computations to a function from computations to computations, which first evaluates a computation and then applies f to the resulting value.

Philip Wadler took the monad to Haskell and showed it can be used to do bad things in a good way. That is, to perform imperative programming language operations in a pure functional language way. Haskell is a pure functional language. In a pure functional language all functions are pure, i.e. functions in which the input-output relation cannot be augmented by things like state, or context, or whatever; the domain – codomain relation is fixed. In a pure functional language you can thus have no side effects, or interactive input / output, etc.

The monad solves this problem by encapsulating ‘imperative operations’ in the M computation. According to Wadler a function

f: a -> M<b>

is then said to map type a onto type b, with a possible additional effect captured by M. The monad takes the ‘imperative operations’ out of the (pure) function application mechanism, and the functional composition chain, and thus saves the pure character of the pure functional language.

Does this talk about functional language has any value for C++? That is debatable. C++ is an imperative language, the idea to improve it by a construct that makes a functional language do imperative operations is preposterous. On the other hand, Bartosz Milewski is convinced that concurrency in C++ would be improved greatly if it would be structured the monadic way (think of the continuations mentioned above). Concurrency is new territory for C++, so we have to investigate this further. I intend to discuss this viewpoint somewhere in this series of articles on monads. First I want to develop a thorough understanding of monad implementations in C++.

Links to Texts on Monads

Who writes or wrote about monads, and what do they think is the merit of monads?

Eugenio Moggi wrote Notions of computation and monads, about. This paper introduces monads as the categorical semantics for computations. It is very hard to read, because full of funny mathematical words that have a very precise meaning which then, of course, hardly anybody knows exactly. According to Moggi, monads allow mathematicians to reason about the equivalence of a very general class of programs.

Philip Wadler introduced monads into Haskell in order to perform ‘imperative operations’ in a decent way. Very readable text.

Mike Vanier wrote an extensive introduction into monads. Hard to read because of much Haskell code. After Philip Wadler he states that monads are key in allowing programmers to write side-effecting programs in pure functional languages.

Bartosz Milewski is a fervent supporter of application of monads to C++, specifically in the context of concurrency. He has written much texts that might make reading Moggi’s text easier. Milewski stresses the idea that std::fucture should be monadic.